The Fourier Transform of Absolute Value of t

This page deals with the absolute value function, |t|. To start, we need to rewrite the function k(t)=|t| and the sum of two other functions (g(t) and h(t)): [Equation 1] Now, since we know what the Fourier Transform of the step function u(t) is, and we also know what the Fourier Transform of a function times t is, we can find the Fourier Transform of the first term in Equation [1]: [Equation 2] Note that we use the following notation for the derivative of the dirac-delta impulse: [Equation 3] We can find the Fourier Transform of the second function (h(t)) in Equation [1] now: [Equation 4] Now, the question is, what is: [Equation 5] To determine this, recall that on the dirac-delta page, we discussed how the dirac-delta impulse can be thought of as the limit of a sequence of functions becoming shorter and higher with n, for n=1,2,3,.... Now, if we look at the derivative of each one of these limiting functions (fn), we get: [Equation 6] Now, lets look at the derivative plus the "reflected" derivative (as in Eq [5]): [Equation 7] Since we know the limit as n goes to infinity of fn is the dirac-delta impulse, we can get an answer to Equation [5]: [Equation 8] Hence, we can get the solution, by combining Equations [2,4,8]: [Equation 9]